The Śrīmad Bhāgavatam is the Purāṇa most completely dedicated to Krishna. It doesn’t give a modern-style definition of the astrological conditions under which Śrī Krishna was born. It gives a more ancient description:
The timing was auspicious. Beauty, peace and goodness was experienced everywhere. The constellation belonging to the one born without birth came into view. The Sun, Moon and stars were very peaceful. All directions and all conditions in nature were beautiful and clean.
– Viz. 10.3.1-5
The “constellation belonging to the one born without birth” is the nakṣatra that belongs to Brahmā (who was born from a flower), i.e. Rohinī.
Description of the conditions on earth and in the sky continue until the 8th text – but without describing things that are recognizable to modern astrologers – mentioning mostly natural omens and conditions. At the end of his commentary on the 8th text, Śrīpad Viśvanātha Cakravartī Ṭhākur’s comments refer the reader to two books for more information about the conditions of Krishna’s birth: Hari-vaṁśa and Kha Manikya.
Hari-vaṁśa notes that Krishna was born in the 8th month of gestation, “to confuse Kaṁsa.”
Kha Manikya (a work that is quite likely lost, I am not able to find more than quotes from it) is an “astrological work” – probably of more recent origin, which describes the birth conditions in more modern astrological terms. Viśvanātha quotes it as saying (according to an English translation by Bhanu Swāmī, edited by Mahānidhi Swāmī):
Moon, Mars, Mercury and Saturn were exalted. Taurus rose. Jupiter was in Pisces, Sun was in Leo, Venus in Libra, Rahu in Scorpio. It was a wednesday at midnight, and the Moon was in Rohinī.
Modern astrologers are either unaware of this depiction – or, are nonplussed by seeing that their calculations cannot make this combination of factors occur at any point in the past. For whatever reason, a large section of modern astrologers don’t use this description of Krishna’s chart but make up their own (for example B.V. Raman and, I believe K.N. Rao).
PS – Going to http://vicdicara.com/videos/, scrolling down to the three videos in the “Gods Horoscope” section, you can find a few hours of comments I made on Krishna’s birth chart, a few years ago.